Cartan coherent configurations

TL;DR

This paper proves that Cartan schemes of simple Lie type groups are 2-separable and have base number 2, enabling efficient recognition algorithms based on their intersection numbers.

Contribution

It establishes 2-separability and base number 2 for Cartan schemes of simple Lie type groups, and introduces a new criterion for 2-separability in homogeneous coherent configurations.

Findings

Cartan schemes of simple Lie type are 2-separable.

The base number of these schemes is 2.

A polynomial-time recognition algorithm is developed for large rank and field size.

Abstract

The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is proved that if $G$ is a simple group of Lie type, then asymptotically, the coherent configuration $\cal X$ is 2-separable, i.e., the array of 2-dimensional intersection numbers determines $\cal X$ up to isomorphism. It is also proved that in this case, the base number of $\cal X$ equals 2. This enables us to construct a polynomial-time algorithm for recognizing the Cartan schemes when the rank of $G$ and order of the underlying field are sufficiently large. One of the key points in the proof of the main results is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.